Page 46 - Instant notes

Physical Chemistry 32

Reflections

For a crystalline solid, the distance between the lattice planes is easily obtained from the
Miller indices, and the unit cell dimensions. The simplest example is that of a primitive
cubic unit cell, for which the distance between planes, d, is simply given by:
2
2
2
2
2
d =a /(h +k +l )
where h, k, and l are the Miller indices and a is the length of the unit cell edge.
Substitution of this relationship into the Bragg condition yields the possible values for θ:
2 1/2
2
2
sinθ=(h +k +l ) λ/2a
2
2
2
Because h, k, and l are whole numbers, the sum (h +k +l ) also yields whole numbers,
2
and because λ and a are fixed quantities, sin θ varies so as to give a regular spacing of
reflections. However, some whole numbers (7, 15, 23, etc.) cannot be formed from the
sum of three squared numbers, and the reflections corresponding to these values of
2
2
2
(h +k +l ) are missing from the series. If λ/2a is denoted A, then the values of sinθ for a
simple cubic lattice are given by: A/√1, A/√2, A/√3, A/√4, A/√5, A/√6, A/√8, A/√9, A/√10,
A/√11, etc. It is therefore possible to identify a primitive cubic unit cell from both the
regularity of the spacings in the X-ray diffraction pattern, and the absence of certain
forbidden lines.
Other unit cells yield further types of missing lines, known as systematic absences.
Simple geometric arguments show that the following conditions apply to a cubic unit
cell:
Allowed reflections
Primitive cubic unit cell all h+k+l
Body centered cubic unit cell h+k+l=even
Face centered cubic unit cell h+k+l=all even or all odd
Similar, but increasingly complex, rules apply to other unit cells and identification of the
systematic absences allows the unit cell to be classified.

Powder crystallography

When a single crystal is illuminated with radiation, reflections are only observed when
one of the lattice planes is at an angle which satisfies the Bragg condition. In the powder
diffraction method, the crystalline sample is ground into a powder, so that it effectively
contains crystals which are oriented at every possible angle to the incident beam. In this
way, the Bragg condition for every lattice plane is simultaneously fulfilled, and
reflections are seen at all allowable values of θ relative to the incident beam (Fig. 3).